lnopsubmuli - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > lnopsubmuli		Structured version Visualization version GIF version

Theorem lnopsubmuli 28834

Description: Subtraction/product property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
lnopl.1	⊢ 𝑇 ∈ LinOp

**Assertion**
Ref	Expression
lnopsubmuli	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))

**Proof of Theorem lnopsubmuli**
Step	Hyp	Ref	Expression
1		hvmulcl 27870	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2		lnopl.1	. . . . . 6 ⊢ 𝑇 ∈ LinOp
3	2	lnopsubi 28833	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
4	1, 3	sylan2 491	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
5	4	3impb 1260	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
6	5	3com12 1269	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
7	2	lnopmuli 28831	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐶)) = (𝐴 ·_ℎ (𝑇‘𝐶)))
8	7	oveq2d 6666	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))
9	8	3adant2 1080	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))
10	6, 9	eqtrd 2656	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℋchil 27776 ·_ℎ csm 27778 −_ℎ cmv 27782 LinOpclo 27804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-hilex 27856 ax-hfvadd 27857 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvdistr2 27866 ax-hvmul0 27867

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-hvsub 27828 df-lnop 28700

This theorem is referenced by: lnopeq0lem1 28864 lnophmlem2 28876

W3C validator